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On \(\alpha\)-convex operators. (English) Zbl 1094.47047

Let \(E\) be a Banach space which is partially ordered by a cone \(P\) in \(E\). Let \(P^0 = \{x \in P: x\) is an interior point of \(P\}\). The cone \(P\) is called solid if its interior \(P^0\) is nonempty. If \(y-x \in P^0\), then this is denoted by \(x << y\). Let \(\alpha \in \mathbb{R}\) and let \(A\) be a positive mapping on \(P^0\). Then \(A\) is called \(\alpha\)-convex iff \(t^\alpha A x \leq A (tx)\) for all \(x \in P^0\) and \(t \in (0, 1]\). The authors study \(\alpha\)-convex operators \((\alpha > 1)\) and show the following fixed point theorems.
Theorem. Let \(E\) be a real Banach space, \(P\) a normal, solid cone, and \(\alpha > 1\). Let \(A: P \rightarrow P\) be an increasing \(\alpha\)-convex operator which satisfies (i) there exist \(u_0, v_0\) in \(P^0\) such that \(0 << A u_0 \leq u_0 < v_0 \leq A v_0\); (ii) there exists a linear operator \(L: E \rightarrow E\) which has an increasing inverse \(L^{-1}\) such that \(Ay - Ax \leq L (y - x)\) for all \(y \geq x \geq 0\). Then \(A\) has a unique fixed point in \([u_0, v_0]\).
Given \(0< h\) in \(E\), let \(P_h = \{x \in E : \exists \lambda (x), \mu (x) > 0\) such that \(\lambda (x) h \leq x \leq \mu (x) h\}\). Theorem. Let \(E\) be a real Banach space and \(P\) be a normal cone in \(E\). Let \(A: P_h \rightarrow P_h\) be an increasing \(\alpha\)-convex operator, \(\alpha > 1\). Assume also that there exists a nonempty totally ordered set \(S \subset P_h \) such that (i) \(\lambda S \subset S (\lambda \in (0,1)\), \(AS = S\); (ii) \(v_0 \leq A v_0\) for some \(v_0 \in S\). Then \(A\) has a unique fixed point in \(S\).
They authors give one application to an integral equation.

MSC:

47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
47H10 Fixed-point theorems
47N20 Applications of operator theory to differential and integral equations
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References:

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